James H. Davenport, Fritz Kr�ckeberg,Ramon E. Moore, Siegfried M. Rump (editors):

Symbolic, Algebraic and ValidatedNumerical Computation

Dagstuhl-Seminar-Report; 4303.08.�07.08.92 (9232)

ISSN 0940-1121

Copyright © 1992 by IBFI GmbH, Schloß Dagstuhl, W-6648 Wadern, GermanyTeI.: +49-6871 - 2458

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Seminar Report 9232

Symbolic, Algebraic and Validated Numerical ComputationAugust 3 - 7, 1992

OVERVIEW

The �rst Dagstuhl Seminar on Symbolic, Algebraic and Validated Numerical Computation was

organized by J. H. Davenport, Bath, F. Kriickeberg, Bonn, R. E. Moore, Columbus, and S. M.Rump, Hamburg. It brought together 32 participants from 5 countries, 7 participants came fromoverseas.

The 26 talks covered a wide range of topics of the three areas Computer Algebra, ValidatedComputation, and Numerical Computation. The aim of the seminar was to bring togetherexperts of those three areas to discuss common interests.

Both Computer Algebra and Validated Computation aim on computing correct results on thecomputer. Here correct is to be understand in a mathematical sense including all model,

discretization and rounding errors. Both approaches can bene�t from Numerical Computationby validating an error bound for an approximation.

In the talks we saw some algorithms with result veri�cation for �nite dimensional as wellas infinite dimensional problems, a promising global optimization algorithm, an interestingapproach to analyze the sensitivity of algebraic problems, and hybrid algorithms combiningtwo or even three of the main areas of the conference. Moreover, we saw a number of practical

applications.

Everybody enjoyed the very pleasant atmosphere, the excellent food and the surroundingsinviting to intensive discussions and recreational hiking.

We would like to express our thanks to all who contributed to the conference and to the

administration of the Dagstuhl center for their excellent job.

Contents

�OLIVER ABERTHFinding where n functions of 72 variables are simultaneously zero

GÖTZ ALEFELD

The Newton-Kantorowif: Theorem and the Veri�cation of Solutions

(together with N. Gienger and F1. Potra)

H. BEHNKE

Bracketing Frequencies of Vibrations of Turbine Blades in Large Turbo-Machines

GEORGE CORLISS

Generating Derivative Codes from Fortran Programs

J. H. DAVENPORT

Introduction to Computer Algebra / Some Symbolic Numeric Case Studies

DAVID M. GAY

Is Exploiting Partially Separable Structure Worthwhile?

K.-U. J AHN

Validated Computation with Sets of Hyperrectangles

CHRISTIAN JANSSON

A Validation Method for Global Optimization Problems

WERNER KRANDICK

High Precision Calculation of Polynomial Complex Roots

FRITZ KRÜCKEBERG

The general Structure of integrated Symbolic, Algebraic and Validated Numerical Computa-tion

WELDON A. LODWICK

Three Applications of Interval Methods for Computing Feasible Region of Nonlinear Systems

RUDOLF LOHNER

Validated Computations for Ordinary Differential Equations

RUDOLPH LORENTZ

A Hybrid Reduce-Fortran Finite Element Test Environment

(together with M. Müller)

DAVID W. MATULA _

Standards and Algorithm Design for Floating Point Arithmetic

GÜNTER MAYER

Two-Stage Interval Iterative Methods

RAMON MOORE

Parameter Identi�cation with Bounded Error Data

MARKUS MÜLLER

A Hybrid Reduce-Fortran Finite Element Test Environment

(together with R. Lorentz)

M.T. NAKAO

Numerical Veri�cations of Solutions of Partial Differential Equations

ARNOLD NEUMAIER

Con�dence regions, ellipsoid arithmetic, and the wrapping effect

M. POHST

On Computing in Algebraic Number Fields

SIEGFRIED RUMP

Symbolic� Algebraic and Validated Computation

MARK SCHAEFER

Precise Optimization using Range Arithmetic

H.J. STETTER

Sensitivity Analysis of Algebraic Algorithms

BERND TIBKEN

Simulation of Uncertain Discrete Systems

PETER WALERIUS

High Precision and Veri�ed Computations of Pole Assignment

MICHAEL A. WOLFE

On Certain Computable Tests and Componentwise Error Bounds

Abstracts

Finding where n functions of n variables are simultaneously zero

by Oliver Aberth

First, a description was given of the goals and problems attempted of precise numerical analysis.Problems must �rst be stated so that they are effectively solvable, that is, have no intrinsicdif�culties. The problem of solving AX = B, where A is an n >< n matrix of constants, B is a

column vector of constants, and X is a column vector of unknowns, was used as an example and

converted int.o several suitable problems for precise numerical analysis.

Next the speci�c problem of �nding where n functions of n variables are simultaneously zero

was discussed. When n = 1, the sign change of f(a) and f(b) is helpful in detecting when a zeroin [(1, b] is present. This can be generalized to the topological degree modul for general n. Thereis an interval arithmetic method of evaluating this for general 72, and the method can be usedto �nd the zeros of the n functions.

The Newton-Kantorowié Theorem and the Veri�cation of Solutions

by G. Alefeld

Using error bounds which follow from the Newton-Kantorowié Theorem we device a method for

bounding approximate solutions of nonlinear systems. From the last two iterates of the �oatingpoint Newton�s-method we construct an interval�vector which is used to test for a solution. The

idea has proved to be very successful.

Bracketing Frequencies of Vibrations of Turbine Blades in Large Turbo-Machines

by Henning Behnke

The natural bending vibrations of a free standing blade are concidered. The mathematicalmodel describing this problem is an eigenvalue problem with a system of ordinary differentialequations of fourth order. The eigenvalues depend on a real parameter Q, the angular velocity. Itis shown, how bounds for the eigenvalue�curves can be computed by means of the Rayhigh-Ritzand N.J. Lehmann procedures. Rounding errors are controlled rigorously by the use of intervalarithmetic.

Generating Derivative Codes from Fortran Programs

by George F. Corliss

The numerical methods employed in the solution of many scienti�c computing problems requirethe computation of derivatives ofa function f : IR" �> IRA�. ADIFOR (Automatic Differentiationin FORtran) is a source transformation tool that accepts Fortran 77 code for the computationof f(x) and writes portable Fortran 77 code for the computation of J (1)5 � where J is the matrix

of first derivatives Jkz = %�£5, and S is a matrix with n rows. T he ADIFOR interface is very&#39;1

�exible. It allows the computation of the J acobian itself (5 = mm), the product of the J acobianwith a vector y (S = y). or the exploitation of known structure of the Jacobian. ADIFOR is

the result of close cooperation between compiler writers and numerical analysists to target real-life optimization and ODE problems. We introduce the principles behind ADIFOR, outline itsfunctionality, and give examples of its use.

Some Symbolic-Numeric Case Studies

by J. H. Davenport

We present two examples of hybrid symbolic and numeric computation.

Stability Analysis

We take the problem of Dr. Tibken: the dynamical system

1&#39;1 Z P1 -112 x113 P2 P1 I2 �

where 121,192 6 [0.4, 0.5]. We can compute many iterations of this process symbolically, then useIRENA (the Interface between REduce and N Ag), to call a numeric minimisation routine tofind the (local) minimum of the 17,- across the range of the pj. This con�rms the fact that theprocess actually converges.

Enzyme Kinetics

This was a presentation of a lengthy calculation, using the methods of Gröbner bases, numericalminimisation and least-squares�tting to estimate parameter values for a reasonably complexbiochemical reaction from experimental data. This can be seen as one example of the problemconsidered by Prof. Moore, of �nding parameter regions in the presence of uncertain data, thoughwe did not approach the problem in precisely that way.

Introduction to Computer Algebra

by J .H. Davenport

In this talk, we survey computer algebra and its part in the wider task of �computation�, notingthat it is only since the advent of digital computers that �computation� has become identi�ed inthe minds of most with �numeric computation�. Computer algebra has the following possibilitiesof interacting with numeric computation.

o It may replace numeric computation. A classic example of this is some extremely complexquadruple integrals related to on-chip capacitance, which were quite difficult to evaluatenumerically, but which needed to be evaluated very often. A symbolic solution, albeit 400lines long, was found to be much preferable, since evaluation was faster and more accurate.

0 It may augment numerical computation, by various techniques which can always be usedby hand, but may become too tedious in practice. These include symbolic differentiation;series expansion (generally combined with symbolic cancellation of dominant terms);expression rearrangement; and, particularly relevant to the theme of this conference,symbolic tracking of the relationship between different interval-valued variables.

o It may be used to analyse numerical computation. One particular area that is drawing alot of attention at the moment is the stability analysis of difference schemes. Computeralgebra can also be used to produce closed-form solutions to compare with numerical ones,or to produce test data or high-precision comparison data for conventional or intervalmethods.

Is Exploiting Partially Separable Structure Worthwhile?

by David M. Gay

As Griewank and Toint have pointed out, many optimization problems have a partially separableobjective function, i.e., one of the form f(:1:) = Z f.-(U,-9:), where a: 6 IR" and U, is a real 12,- >< n

iep

matrix with n,- << n. Backward Automatic Differentiation (AD) applied either to dTV f or toV,,<,o�(0), where c,o(a) = f(x + ad) [see Christianson�s recent paper in SMA J. Num. Anal.] candeliver (V2 f )d (i.e., Hessian of f times vector d) in a number of operations proportial to thoseneeded to compute f. How does computing (V2 f )d compare with the conventional technique ofcomputing a quasi-Newton approximation H,- to V2f,- and computing E(U,-TH,-U,-)d, both in a

itruncated Newton scheme?

Another way to exploit partially separable structure with AD is to compute the n,- columns ofV2f,~, then use either the explicit Vzf = Z U,TV2f,-U; or (V2f)d = 2 U,-TV2f,-U,-d. How do

�l �I

these alternatives compare with those above?Initial computational results comparing just the �nite difference [V f(a: + hd) � V/h witha backward AD computation of (V2 f )d show the �nite difference to be somewhat faster. I hadhoped by now to have more computational experience in this area, but have been waylaid bywork on a forthcoming book on AMPL (a modeling language for mathematical programming,joint work with Bob Fourer and Brian Kernighan). This itself is a project relevant to thismeeting, as the AMPL translator converts an algebraic description of an optimization probleminto a symbolic representation � expression DAGs � that other programs, e.g., solvers, can

manipulate.

Validated Computation with Sets of Hyperrectangles

by Karl-Udo J ahn

Due to the special shape of axis-parallel intervals during enclosure processes for solution setsoverestimations may occur. That�s why more general enclosure sets are of interest. These setsshould have most of the useful properties of intervals (reprensentable by a few parameters;Minkowski operations, outwardly directed roundings, intersection as well as interval hull areeffective performable; Hansdorff and maximum distances are computable, extensions of functionscan be de�ned), and they should be useful to avoid the wrapping effect, to allow furtheroperations and to approximate arbitrary closed bounded sets as close as one wishes. It canbe shown that �nite sets of intervals resp. their unions ful�ll the above-mentioned requirements.Furthermore, Banach-like �xed point theorems are available.

A Validation Method for Global Optimization Problems

by Christian Jansson

A method for the global optimization problem with simple bound constraints is presented.The method is based on the tools of interval arithmetic and uses a special branch-and-boundtechnique in connection with a descent algorithm. Besides the calculation of approximations ofthe global minimum value and global minimum points in addition guaranteed lower and upperbounds of the global minimum value are computed. Derivatives of the function are not required.Numerical results on a large set of test functions are presented.

High Precision Calculation of Polynomial Complex Roots

by Werner Krandick

An algorithm is presented which calculates polynomial complex roots to speci�ed precision.The method is infallible, i.e. unlike numerical algorithms. It will always terminate with correctresults. Its average computing time seems to be cubic in the degree of the polynomial, and linearin the length of the polynomial coefficients. Preliminary experiments indicate a quadratic rate

of convergence.

Applying the �Principle of the argument� from complex analysis a winding number computationis used to determine the number of polynomial roots in a rectangle. This method is appliedunder a bisection scheme in order to compute isolating rectangles for the complex roots of apolynomial, i.e. rectangles which contain exactly one root. Once a sufficiently small isolatingrectangle has been found, a geometrical construction using tangents to certain algebraic curvesguesses a (usually very small) subrectangle which is likely to contain the root. The (infallible)winding number computation is used to validate the guess; if the guess proves incorrect, bisectionis used to obtain a subrectangle. This process continues until isolating rectangles for all rootshave been found and re�ned as desired.

The general Structure of integrated Symbolic, Algebraic andValidated Numerical Computation

by Fritz Kriickeberg

By an integration of Symbolic Computation, Computer Algebra and Numerical Computation wehave the possibility of producing a new generation of powerful methods. These new methods canhelp to ful�ll the needs of users, engineers and also the needs for veri�cation and correctness. Thegeneral strategy: To get a new generation of methods by synthesis of Numerical Mathematics andPure Mathematics by the computer. The paradigm of future oriented computing is to realize:Veri�cations, Flexibility and Efficiency. The general structure consists of three (or more) levelswith a process-control backward and forward:

Symbolic Computation Computer Algebra Systems nNumerical Methods Numerical Methods and Procedures -

(1) (Interval)�Arithmetic with variable precision mT} The Levels 1T The Procedures if The Control

What we need is a deeper understanding (a theory) of

§/

(3) how to �nd and select the best formal representation,

(2) how to �nd the best numeric.al method and how to control their parameters,(1) how to control the word-length and how to select the arithmetic representation

Three Applications of Interval Methods for Computing Feasible Regions of NonlinearSystems

by Weldon A. Lodwick

Interval methods for computing feasible regions of nonlinear systems associated with threeapplications were discussed. Work in progress and some results were presented. The applicationswere in the areas of processing models, such as re�neries, radiation therapy of cancer tumors

and constraint logic programming.

Validated Computations for Ordinary Differential Equations

by Rudolf Lohner

For different kinds of problems with ordinary differential equations methods are presented whichcan prove the existence of solutions and verify bounds by numerical computation with intervalarithmetic. The basic type of problems are initial value problems (ivp) where Taylor expansionsare used and their remainder terms are bounded by intervals. Boundary value problems (bvp)are reduced to ivp and �nite dimensional equations by means of a multiple shooting methodwhich is also executed in interval arithmetic. Also eigenvalue problems free boundary valueproblems and others can be handled by transforming them to standard bvp on a �xed interval.The methods have been programmed in PASCAL-XSC and examples are presented which showthe applicability of the method (including chaotic solutions of the Lorenz equations and mildlystiff bvp).

A Hybrid Reduce-Fortran Finite Element Test Environment

by R.A. Lorentz & M. Müller

The basic idea of the above mentioned test environment is to allow one to determine whether

or not a given multivariate interpolation (by multivariate polynomials) can be used for thenumerical solution of self-adjoint partial differential equations via the �nite element method.

In particular, the user should only be required to formulate the interpolation after which the

program carries out all the computations by itself.

The class of interpolating functionals which are supported include not only the usual function

10

values, derivatives at points and normal derivatives at points, but also non-standard functionals.For example, mean values of functions or their derivatives over edges or regions, which have been

used by Kergin and Hakopicen, are also included. The user of this test environment is, however,not restricted to using the implemented functionals since the environment has been concievedas an open system. It is extremely easy not only to include user-de�ned functionals but, eg, tochange the norms used to evaluate the error.

The computer algebra program REDUCE is used to determine whether the set of functionals

speci�ed together with the space of polynomials to be interpolated from de�ne a regularinterpolation. It also determines the form functions of interpolation and calculates the elementstiffness matrix of the �nite element discretization. It also generates Fortran code for these

quantities and passes it to a numerical �nite element solver. Finally, the quality of the

discretization, which includes the size of the errors and the convergence rate, is determined

and if desired, graphics are generated.

The test environment runs on all platforms which support REDUCE and Fortran. I�. can beobtained from either of the authors. &#39;

Standards and Algorithm Design for Floating Point Arithmetic

by David W. Matula

We describe how the IEEE standard provided closure and uniqueness to �oating point arithmeticand established a more meaningful hierarchical precision environment. The standard adressedthe basic single operations :l:, ><� /� \/, but did not addres more complicated operations, such asthe inner product. Following the commercially appropriate model established by the standardthat error monitoring and well de�nedness will be added to available hardware provided the timeand hardware resources required are only marginally greater than for unvalidated computation,we attempt to provide suitable validation for the inner product.

We characterize and describe a threshold for catastrophic concellation in an inner product. Wedescribe variations in in�nitely precise rounding including the existance of a back up mode,i.e. �round to nearest or otherwise round down� (RND). We introduce the notions of stickyaccumulation in higher precision. For the particular example of quad (128 bit) stickyaccumulation of double precision (64 bit) �oating point numbers, we show a cost effective designfor inner product that can detect catastrophic cancellation, otherwise in its absence will yielda validated precise RND rounding. We �nally discuss a software model of the process which isfairly ef�cient given only the extra hardware instruments for low order parts of multiply andadd.

Two-stage Interval Iterative Methods�

by Günter Mayer

Two-stage methods are iterative methods to approximate solutions of linear systems A1? = bwith A E IHM", b 6 IR". They are based on two splittings A = M � N and M = F � G whichare used to construct the iteration

11

.?°::1+k

for i z: 1 to 3A solve F? := G;&#39;fi"1 + Nrk + b

1,k+1 :: "s(k)

where .s(k) E IN. Two-stage interval methods are de�ned analogously for n X n interval matrices

[A] and interval splittings [A] = :• � • and [NI] : [F] � [G], where the loop above is replacedby

(f? == 19-4([Fl»[G&#39;ll?l"" + [Nl[rl�° + [5])-

Here, IGA(., �Á denotes the vector which is generated by the interval Gaussian algorithm. Resultsare given for s(k) E .3 = const (stationary method) and for s(k) being allowed to vary with theiteration index k (instationary method). These results refer to the global convergence of themethod, to the speed of convergence, to the quality of enclosure of the solution set L = {r |Ar = b, A E [A], b E [b] } E IIR") and to the nonsingularity of A E

*l joint work with Andreas Frommer

Parameter Identi�cation with Bounded Error Data

by Ramon Moore

The problem is to �nd the set of all parameter vectors for which a model agrees with data to

within given bounds on errors in the data.

Using interval methods, the problem is now solved.

Numerical Veri�cations of Solutions of Partial Differential Equations

by Mitsuhiro T. Nakao

We consider a numerical technique to verify the solutions for nonlinear elliptic boundary valueproblems. Using a �nite element solution and explicit error estimates for certain simple linearproblems, we construct, on the computer, a set of functions which satis�es the condition of

Schauder�s or other �xed point theorem in the in�nite dimensional space. In order to obtainsuch a set. in numerical settings, the method of computable error estimation plays an essentialrole. Some numerical examples are presented.

The veri�cation principle provided can also be applied for evolutional problems.

Con�dence regions, ellipsoid arithmetic, and the Wrapping effect

by Arnold Neumaier

The wrapping effect is one of the main reasons that applications of interval arithmetic to theenclosure of dynamical systems is difficult. In this talk, the various sources of wrapping areanalyzed. A new method for reducing the wrapping effect is proposed, based on ellipsoidarithmetic.

12

As an application, it is shown how one can propagate ellipsoidal con�dence regions for stochastic

parameters through arithmetical expressions without reducing the con�dence level.

On computing in algebraic number �elds

by Michael E. Pohst

We discuss some problems related to doing arithmetic in an algebraic number �eld F. The(integral) elements of F under consideration are supposed to belong to an order R of F. Thenaddition, subtraction and multiplication are immediate via the presentation of the elements ofR by some �xed Z-basis. However, division in R is already rather complicated and we discussseveral distinct methods.

A major topic of computational algebraic number theory is the computation of the unit groupU (R) of R. We sketch a new method which essentially makes use of two algorithms from thegeometry of numbers, namely

(i) LLL-reduction of lattice bases,

(ii) computation of short(est) vectors in a lattice.

Instead with varying lattices given by bases we use only the corresponding Gram matrices andadd weights if appropriate. While accuracy is not required for some applications (where theresult can be easily checked for correctness) we need to use validated numerical computationsfor proving that certain regions do not contain lattice points. �

Symbolic, Algebraic and Validated Computation

by Siegfried M. Rump

Both Computer Algebra and Validated Computations aim on computing correct results on thecomputer, correctness to be understand in a mathematical sense including all model,discretization and rounding errors. Correct results may be. given in terms of exact results(rational numbers, algebraic numbers etc.) or enclosing intervals with veri�ed existence andpossibly uniqueness of the solution within the computed bounds. There are many applicationswhere the latter way of giving results is sufficient. This is a joint area both computer Algebraand Validation Algorithms may focus on.

Precise Optimization Using Range Arithmetic

by Mark J. Schaefer

We recently developed an implementation of range arithmetic in C++ to serve as the basis forwriting numerical programs which compute reliable answers. This arithmetic is more complexthan �oating�point arithmetic and is a form of interval arithmetic. My talk will provide a briefintroduction to range arithmetic and will focus on the adaptation of a Levenberg�Marquardtsearch algorithm for local minima to this arithmetic in order to guarantee convergence andgenerate correct results. A few numerical examples will be discussed as well.

13

Sensitivity Analysis of Algebraic Algorithms

by Hans J. Stetter

Nearly all mathematical problems permit degenerations, often of various degrees (rank de�ciency,special positions, etc.). Solution algorithms for such problems should recognize near-degeneratesituations and tell us about it: In applications with not fully accurate data we have to assume

that the degeneracy exists, and in any case we should treat the problem as a perturbation ofthe degenerate one for better condition.

In good numerical �oating-point algorithms, near degeneracies will show up automaticallybecause a potential loss of accuracy must be checked. If rational arithmetic ( without wordlengthrestriction) is used in algebraic algorithms the problem appears either as precisely degenerateor not at all, at least in present-day implementations is Computer Algebra systems.

In our presentation, we demonstrate that all algebraic algorithms should check for nearbydegeneracies. In many cases this can be achieved (with appropriate scaling) by droppingsuf�ciently small quantities and doing an a posteriori analysis of the effect of these perturbations.After the precise algorithm has been thus modi�ed, it can also be turned into �oating-pointversions by inclusion of the round-off effects into the perturbation analysis.

The following problem areas have been considered:

�� Multivariate interpolation

�� Symbolic integration of rational functions

�- G.c.d. of two univariate polynomials

�- Joint zeros of multivariate polynomials

It is claimed that algebraic algorithms which have been. thus enhanced are more suitable forScienti�c Computing.

Simulation of Uncertain Discrete Systems

by Bernd Tibken and Eberhard P. Hofer

The simulation problem for discrete time systems a:(k + 1) = f(a:(k), u(k), p, t); y(k)= h(:z:(k), u(k), p, k) is discussed in the uncertain parameter p and initial condition :1:(0) setting.The naive use of interval arithmetic is compared to a sophisticated approach which makes use

of the global optimization algorithm by Hansen. Componentwise inclusions for the state a: andthe output y of the system are computed. The results will be published in the proceedings ofan NASA conference held in June �92. A copy is available from the authors.

14

High precision and veri�ed Computations of Pole Assignment

by Peter Walerius

The pole assignment, carried out by state variable feedback is a method for the synthesis ofcontrol systems. The presented method introduced here is based on an explicit formula for state

variable feedback design in case of multivariable system. This formula yields from the theory ofthe Complete Modal Synthesis, which is transformed into a system of nonlinear equations. Usingan inclusion procudure for the solution of nonlinear systems, the computation of the feedback

matrix the pole placement problem can be carried out with high precision and veri�cation.

On Certain Computable Tests and Componentwise Error Bounds

by Shen Zuhe & M. A. Wolfe

The use of interval slope arithmetic in Pandian�s existence test and componentwise error boundsfor solutions of nonlinear systems f : D Q IRN �-> IRN with f 6 C1(D) (SIAM J.N.A. &#39;22 (1985))is investigated. The results which are obtained are related to the results of Moore & Kioustelidis(SIAM J.N.A. 17 (1980)) and of Shen & Neumaier (Computing 40(1988)).

15

Dagstuhl-Seminar 9232: List of Participants

Oliver Aberth Karl-Udo JahnTexas A&M University Technische Hochschule LeipzigDept. of Mathematics Sektion Mathematik/Informatik

i College Station TX 77843-3368 Karl-Liebknecht-Stra�e 132USA O-7030 Leipzigtel.: +1 -409-693-55 25 Germany

tel. : +49-341 -3928-482

Götz AlefeidUniversität Karlsruhe Christian JanssonInstitut für Angewandte Mathematik TU Hamburg-HarburgKaiserstr 12 Technische Informatik IIIW-7500 Karlsruhe 1 Eißendorfer Str. 38Germany W�2000 Hamburg 90na.aIefeId@na-net.standford.edu Germanytel.: +721-608-20 60 jansson@tu-harburg.dbp.de

teI.: +40-7718-29 23

Henning BehnkeTU Clausthal Fritz Kriickeberg/IInstitut für Mathematik Gesellschaft für athematik undErzstraße 1 Datenverarbeitung mbHW-3392 CIausthaI-Zellerfeld Schloß BirlinghovenGermany Postfach 1316mahb@IBM.RZ.TU-CLAUSTHAL.DE W�5205 St. Augustin 1tel.: +5321-72-31 83 Germany

krue@gmdzi.deteI.: +2241-14�23 35 und 23 34

George CorlissDepartment of MathematicsMilwaukee WI 53233 Werner KrandickUSA Johannes Kepler Universitätgeorgec@bris.mscs.mu.edu Institut für Mathematiktel.: +1 -414-288-6599 RISC

A-4040 LinzAustria

James H. Davenport krandick@risc.uni-Iinz.ac.atUniversity of Bath tel.: +43-7236-3231-44School of MathematicsClaverton DownBath BA2 7AY Weldon A. LodwickGreat Britain University of Colorado at Denverjfd@maths.bath.ac.uk Dept. of Mathematicstel.: +44-225-826181 Campus Box 170

1100 Fourteenth StreetDenver CO 80217-3364

David M. Gay USAAT&T Bell Laboratories wIodwick@copper.denver.colorado.eduRoom 2C-463 tel.: 1-303-556-8442600 Mountain AvenueMurray Hill NJ 07974-0636USA

dmg@research.att.comtel.: +1 �908�582-56 23

Rudolf LohnerUniversität KarlsruheInst. f. Angewandte MathematikKaiserstraße 12W-7500 Karlsruhe 1

Germanyae34@iamk4508.mathematik.uni�karlsruhe.deteI.: +721 -608-42 73

Rudolph LorentzGesellschaft für Mathematik undDatenverarbeitung mbHSchloß BirlinghovenPostfach 12 40W-5205 St. Augustin 1Germanylorentz@.gmd.deteI.: +2241 -33 21 18

Günter LudykUniversität BremenInstitut für AutomatisierungstechnikPostfach 33 04 40W-2800 Bremen 33

GermanyteI.: +421�27 23 84

Markus MüllerGMD Schloß Bimghoven F1TInstitut für Methodische GrundlagenPostfach 12 40W-5205 Sankt Augustin 1

Germany gmap51@f1ibmsv.gmd.deteI.: +2241-14 27 54

David W. MatulaSouthern Methodist UniversiyDepart. of Computer Science & EmgneeringDallas TX 75275USAmatula@seas.smu.eduteI.: +1-214-692 3089

Günter MayerUniversität KarlsruheInstitut für Angewandte MathematikKaiserstr 12W-7500 Karlsruhe 1

Germanyae09@dkauni2.bitnetteI.: +721/608-26 83

Ramon MooreOhio State UniversityDepart. of Computer 8. Information Science2036 Neil AvenueColumbus OH 43210-1277USAmoore-r@cis.ohio-state.eduteI.: +1 -61 4-846-0903

Mitsuhiro Nakao

Kyushu University 33Dept. of MathematicsPostal No. 812Fukuoka

JapanteI.: +81-O92-641-1101-43 74

Arnold NeumaierUniversität FreiburgInstitut für Angewandte MathematikHermann Herder Str. 10W-7800 Freiburg i. Breisgau

Germany neum@sun1.ruf.uni-freiburg.deteI.: +761�2 03 30 65

Michael E. PohstUniversität DüsseldorfMathematisches InstitutUniversitätsstr. 1W-4000 Düsseldorf 1

Germany pohst@ze8.rzuni-duesseldorfdeteI.: +211�311-21 88

Georg RexUniversität LeipzigSektion MathematikAugustusplatz 100-7010 Leipzig

Germany rex@mathematik.uni-Ieipzig.dbp.deteI.: +719-23 17/18

Siegfried M. Rump

TU Hamburg-HarburgTechnische nformatik IIIEisendorfer Str. 38W-2100 Hamburg 90Germany

Mark J. Schaefer

Universität Tübingen

WiIhelm-Schickhard�Institut Sand 13W-7400 TübingenGermany

Hans J. StetterTU WienInstitut für Angewandte undNumerische MathematikWiedner Hauptstr. 8/1 OA-1040 Wien

Austria stetter@uranus.tuwien.ac.atteI.: +43-1-58801-54 08

Bernd TibkenUniversität Ulm

Abteilung Meß-Regel- u. MikrotechnikPostfach 40 66 W-7900 Ulm

Germany+731-502-41 92

Peter WaleriusUniversität BremenInstitut für AutomatisierungstechnikKufsteiner Str.W-2800 Bremen 33

GermanyteI.: 0421-218-3490

Michael A. WolfeUniversity o1 St. AndrewsNorth HaughSt. Andrews KY16 9SSGreat Britainmaw@st�andrews.ac.ukteI.: +44-334-761 61 -81 75

P. Klint, T. Reps, G. Snelting (editors):Programming Environments; Dagstuhl-Seminar-Report; 34; 9.3.-13.3.92 (9211)

H.-D. Ehrich, J.A. Goguen, A. Sernadas (editors):Foundations of Information Systems Specification and Design; Dagstuhl-Seminar-Report; 35;16.3.-19.3.9 (9212)

W. Damm, Ch. Hankin, J. Hughes (editors):Functional Languages:Compiler Technology and Parallelism; Dagstuhl-Seminar-Report; 36; 23.3.-27.3.92 (9213)

Th. Beth, W. Diffie, G.J. Simmons (editors):System Security; Dagstuhl-Seminar-Report; 37; 30.3.-3.4.92 (9214)

C.A. Ellis, M. Jarke (editors):Distributed Cooperation in Integrated Information Systems; Dagstuhl-Seminar-Fleport; 38; 5.4.-9.4.92 (9215)

J. Buchmann, H. Niederreiter, A.M. Odlyzko, H.G. Zimmer (editors):Algorithms and Number Theory, Dagstuhl-Seminar-Fleport; 39; 22.06.-26.06.92 (9226)

E. Borger, Y. Gurevich, H. Kleine-Btining, MM. Richter (editors):Computer Science Logic, Dagstuhl-Seminar-Report; 40; 13.07.-17.07.92 (9229)

J. von zur Gathen, M. Karpinski, D. Kozen (editors):Algebraic Complexity and Parallelism, Dagstuhl-Seminar-Report; 41; 20.07.-24.07.92 (9230)

F. Baader, J. Siekmann, W. Snyder (editors):6th lntemational Workshop on Unification, Dagstuhl-Seminar-Fteport; 42; 29.07.-31.07.92 (9231)

J.W. Davenport, F. Kr�ckeberg, RE. Moore, S. Rump (editors): _Symbolic. iilgebraic and validated numerical Computation, Dagstuhl-Seminar-Fleport; 43; 03.08.-07.08.92 (9232)

R. Cohen, R. Kass� C. Paris, W. Wahlster (editors):Third international Workshop on User Modeling (UM�92), Dagstuhl-Seminar-Report; 44; 10.-13.8.92 (9233) -

Ft. Reischuk, D. Uhlig (editors):Complexity and Realization of Boolean Functions, Dagstuhl-Seminar-Report; 45; 24.08.-28.08.92(9235) &#39;

Th. Lengauer, D. Schornburg, M.S. Waterman (editors):Molecular Biointormatics, Dagstuhl-Seminar-Report; 46; 07.09.-11.09.92 (9237)

V.Fl. Basili, H.D. Rombach, R.W. Selby (editors):Experimental Software Engineering Issues, Dagstuhl-Seminar-Report; 47; 14.-18.09.92 (9238)

Y. Dittrich, H. Hastedt, P. Schete (editors):Computer Science and Philosophy, Dagstuhl-Seminar-Fleport; 48; 21 .09.-25.09.92 (9239)

HP. Daley, U. Furbach, K.P. Jantke (editors):Analogical and Inductive Inference 1992 , Dagstuhl-Seminar-Report; 49; 05.10.-09.10.92 (9241)

E. Novak, St. Smale, J.F. Traub (editors): .Algorithms and Corrplexity of Continuous Problems, Dagstuhl-Seminar-Report; 50; 12.10.-16.10.92 (9242)

J. Encarnacao, J. Foley (editors):Multimedia - System Architectures and Applications, Dagstuhl-Seminar-Report; 51; 02.11.-06.1 1.92 (9245)

F.J. Flammig, J. Staunstrup, G. Zimmermann (editors):Self-Timed Design, DagstuhI-Seminar-Heport; 52; 30.11.-04.12.92 (9249 )

B. Courcelle, H. Ehrig, G. Fiozenberg, H.J. Schneider (editors):Graph-Transformations in Computer Science, Dagstuhl-Seminar-Report; 53; 04.01.-08.01.93(9301)

A. Arnold, L. Priese, R. Vollmar (editors):Automata Theory: Distributed Models, Dagstuhl-Seminar-Report; 54; 11.01 .-15.01 .93 (9302)

W.S. Cellary, K. Vidyasankar , G. Vossen (editors):Versioning in Data Base Management Systems, Dagstuhl-Seminar-Report; 55; 01.02.-05.02.93(9305)

B. Becker, R. Bryant, Ch. Meinel (editors):Computer Aided Design and Test , Dagstuhl-Seminar-Report; 56; 15.02.-19.02.93 (9307)

M. Pinkal, R. Scha� L. Schubert (editors):Semantic Formalisms in Natural Language Processing, Dagstuhl-Seminar-Report; 57; 23.02.-26.02.93 (9308)

H. Bibel, K. Furukawa, M. Stickel (editors):Deduction , Dagstuhl-Seminar-Report; 58; 08.03.-12.03.93 (9310)

H. Alt, B. Chazelle, E. Welzl (editors):Computational Geometry, Dagstuhl-Seminar-Report; 59; 22.03.-26.03.93 (9312)

J. Pustejovsky, H. Kamp (editors):Universals in the Lexicon: At the Intersection of Lexical Semantic Theories, Dagstuh|-Seminar-Report; 60; 29.03.-02.04.93 (9313)

W. Straßer, F. Wahl (editors):Graphics & Robotics, Dagstuhl-Seminar-Report; 61; 19.04.-22.04.93 (9316)

C. Beeri, A. Heuer, G. Saake, S.D. Urban (editors):Formal Aspects o1 Object Base Dynamics � Dagstuhl-Seminar-Report; 62; 26.04.-30.04.93 (9317)

R. Book, E.P.D. Pednauit, D. Wotschke (editors):Descriptional Complexity: A Multidisciplinary Perspective , Dagstuhl-Seminar-Report; 63; 03.05.-07.05.93 (9318)

M. Wirsing, H.-D. Ehrich (editors):Specification and Semantics, Dagstuhl-Seminar-Report; 64; 24.05.-28.05.93 (9321)

M. Droste, Y. Gurevich (editors):Semantics of Programming Languages and Algebra, Dagstuhl-Seminar-Report; 65; 07.06.-1 1.06.93 (9323)

G. Farin, H. Hagen, H. Noltemeier (editors):Geometric Modelling, Dagstuhl-Seminar-Report; 66; 28.06.-02.07.93 (9326)

Ph. Flajolet, R. Kemp, H. Prodinger (editors):"Average-Case�-Analyse von Algorithmen, Dagstuhl-Seminar-Fleport; 67; 12.07.-16.07.93 (9328)

J.W. Gray, A.M. Pitts, K. Sieber (editors):Interactions between Category Theory and Computer Science, Dagstuhl-Seminar-Report; 68;19.07.-23.07.93 (9329)

V. Marek, A. Nerode, P.H. Schmitt (editors):Non-Classical Logics in Computer Science, Dagstuhl-Seminar-Report; 69; 20.09.-24.09.93(9338)

A. Odlyzko, C.P. Schnorr, A. Shamir (editors):Cryptography, Dagstuhl-Seminar-Report; 70; 27.09.-01.10.93 (9339)